Finite Group Databasebeta

← #467328

Subgroups — #467328

Order 512.

1182 conjugacy classes of subgroups.

The subgroup lattice has 557 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1988C211yesmaximal, characteristic, complemented
2C2×#1925C211yesmaximal, characteristic
3#15144C211yesmaximal, characteristic
4#13918C211yesmaximal, characteristic, complemented
5#11384C211yesmaximal, characteristic, complemented
6#11322C211yesmaximal, characteristic, complemented
7#11256C212yesmaximal, complemented
8#11227C212yesmaximal, complemented
9#11076C212yesmaximal
10#11076C211yesmaximal, characteristic
11#11066C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×#102C2211yescharacteristic
17C22×#102C2211yescharacteristic
18C22×#102C2212yes—
19C2×#393—21no—
20C2×#393—21no—
21C2×#390C2211yescharacteristic
22C2×#390C2212yes—
23C2×#389C2212yes—
24C2×#382—21no—
25C2×C4×C4⋊C4C2211yescharacteristic
26#1988—24no—
27#1925—24no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36#1442—22no—
37#1439—22no—
38C4⋊#95C2212yes—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2212yes—
41#1434—21no—
42C4×#95C2211yescharacteristic
43C23×Q8—21no—
44C23×D4C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2311yescharacteristic
48C2×C4⋊Q8—24no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—42no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C22⋊C4—21no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—22no—
63C22×C42—21no—
64C22×C42C2312yes—
65C22×C42C2311yescharacteristic
66C2×#107—22no—
67C2×#107—42no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—42no—
70C2×C4⋊C8—42no—
71C2×C4⋊C8—42no—
72C2×C4⋊C8—42no—
73C2×#102—22no—
74C2×#102—22no—
75C2×#102—22no—
76C2×#102—42no—
77C2×#102—42no—
78C2×#102—42no—
79C2×#102—42no—
80C2×#102Q814yescomplemented
81#393—44no—
82#393—44no—
83#390—24no—
84#390—22no—
85#390—22no—
86#390—42no—
87#389—24no—
88#389—42no—
89#382—44no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4D414yes—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95D411yescharacteristic
101C2×#95D412yes—
102C2×#95D411yescharacteristic
103#340—22no—
104#340—24no—
105#340—24no—
106#340—24no—
107#340—42no—
108#340—42no—
109#336—22no—
110#336—42no—
111C25—21no—
112C22×Q8—44no—
113C22×Q8—42no—
114C22×Q8—42no—
115C22×Q8—42no—
116C22×Q8—82no—
117C22×D4—21no—
118C22×D4—22no—
119C22×D4—21no—
120C22×D4—21no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—21no—
124C22×D4—22no—
125C22×D4—21no—
126C22×D4C2×Q812yes—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4C2×D411yescharacteristic
136C23×C4C2411yescharacteristic
137C22×C8—21no—
138C22×C8—22no—
139C22×C8—21no—
140C22×C8—22no—
141C22×C8—21no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8C2×Q812yes—
154C4⋊Q8—28no—
155C4⋊Q8—48no—
156C4⋊Q8—44no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×Q812yes—
186C2×C4⋊C4#4114yes—
187C2×C4⋊C4#4112yes—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—82no—
198C2×C22⋊C4—82no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—24no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—42no—
209C2×C42—42no—
210C2×C42C2×D412yes—
211C2×C42#4112yes—
212#107—44no—
213C4⋊C8—44no—
214C4⋊C8—48no—
215#102—28no—
216#102—44no—
217#102—48no—
218#95—24no—
219#95—22no—
220#95—42no—
221#95—44no—
222#95—44no—
223#95—44no—
224#95—44no—
225#95—42no—
226#95—44no—
227#95—81no—
228#95D814yes—
229C24—22no—
230C24—41no—
231C24—42no—
232C24—42no—
233C24—41no—
234C24—41no—
235C24—41no—
236C24—42no—
237C24—42no—
238C24—41no—
239C24—41no—
240C24C22×D411yescharacteristic
241C2×Q8—48no—
242C2×Q8—88no—
243C2×Q8—88no—
244C2×Q8—84no—
245C2×Q8—84no—
246C2×D4—22no—
247C2×D4—24no—
248C2×D4—24no—
249C2×D4—22no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—41no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4#12112yes—
315C22×C4#12112yes—
316C22×C4C2×D814yes—
317C22×C4C2×D812yes—
318C22×C4AGL(1,ℤ/8)12yes—
319C22×C4C22×D411yescharacteristic
320C22×C4C22×Q811yescharacteristic
321C22×C4C2×#4111yescharacteristic
322C22×C4C2×#4112yes—
323C22×C4C2×#4111yescharacteristic
324C22×C4H(5,𝔽2)12yes—
325C22×C4H(5,𝔽2)12yes—
326C22×C4H(5,𝔽2)11yescharacteristic
327C22×C42−1+412yes—
328C22×C42−1+411yescharacteristic
329C2×C8—22no—
330C2×C8—24no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—42no—
334C2×C8—41no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—24no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—24no—
353C4⋊C4—48no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—88no—
365C4⋊C4C22⋊Q814yes—
366C22⋊C4—88no—
367C22⋊C4—84no—
368C22⋊C4—88no—
369C22⋊C4—84no—
370C42—22no—
371C42—22no—
372C42—24no—
373C42—42no—
374C42—42no—
375C42—42no—
376C42—42no—
377C42—48no—
378C23—42no—
379C23—42no—
380C23—44no—
381C23—42no—
382C23—44no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23—82no—
387C23—82no—
388C23—82no—
389C23—82no—
390C23—82no—
391C23—81no—
392C23—82no—
393C23—81no—
394C23—81no—
395C23—81no—
396C23—81no—
397C23—82no—
398C23—82no—
399C23—82no—
400C23—82no—
401C23C2×#12111yescharacteristic
402C23#53712yes—
403C23#54611yescharacteristic
404C23#54612yes—
405C23D4×Q811yescharacteristic
406C23C22×D811yescharacteristic
407C23C2×AGL(1,ℤ/8)11yescharacteristic
408C23#57512yes—
409C23#57711yescharacteristic
410C23#57712yes—
411C23#57811yescharacteristic
412Q8—816no—
413D4—42no—
414D4—42no—
415D4—44no—
416D4—81no—
417D4—81no—
418D4—82no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—81no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—81no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—81no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4#45914yes—
474C2×C4C2×C22⋊Q811yescharacteristic
475C2×C4#52511yescharacteristic
476C2×C4#53511yescharacteristic
477C2×C4#53511yescharacteristic
478C2×C4#53611yescharacteristic
479C2×C4#53711yescharacteristic
480C2×C4#54112yes—
481C2×C4#54212yes—
482C2×C4#54212yes—
483C2×C4#54212yes—
484C2×C4#54412yes—
485C2×C4#55112yes—
486C2×C4#55211yescharacteristic
487C2×C4#55412yes—
488C2×C4#55411yescharacteristic
489C2×C4#55912yes—
490C2×C4H(3,𝔽4)12yes—
491C2×C4#56212yes—
492C8—42no—
493C8—81no—
494C8—82no—
495C22—84no—
496C22—84no—
497C22—82no—
498C22—82no—
499C22—82no—
500C22—162no—
501C22—162no—
502C22—161no—
503C22—161no—
504C22—161no—
505C22—161no—
506C22#241111yescharacteristic
507C22#241711yescharacteristic
508C22#262011yescharacteristic
509C22#270412yes—
510C22#271212yes—
511C22C2×#45911yescharacteristic
512C22#304011yescharacteristic
513C22#304311yescharacteristic
514C22#304611yescharacteristic
515C22#324512yes—
516C22#325112yes—
517C22#325812yes—
518C22#326312yes—
519C22#329411yescharacteristic
520C22#329412yes—
521C22#331012yes—
522C22#332112yes—
523C22#332112yes—
524C22#332712yes—
525C22#333011yescharacteristic
526C22Q8×D811yescharacteristic
527C22#338411yescharacteristic
528C22#338812yes—
529C4—24no—
530C4—44no—
531C4—42no—
532C4—44no—
533C4—82no—
534C4—82no—
535C4—84no—
536C4—162no—
537C4—161no—
538C4#258011yescharacteristic
539C4#262811yescharacteristic
540C4#263711yescharacteristic
541C4#275412yes—
542C4#276312yes—
543C4#281911yescharacteristic
544C2—162no—
545C2#1656611yescharacteristic
546C2#2447611yescharacteristic
547C2#2449911yescharacteristic
548C2#2481111yescharacteristic
549C2#2482811yescharacteristic
550C2#2982411yescharacteristic
551C2#2984111yescharacteristic
552C2#3186612yes—
553C2#3189512yes—
554C2#3211012yes—
555C2#3212212yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#1988C2semidirect
4. #13918C2semidirect
5. #11384C2semidirect
6. #11322C2semidirect
7. #11256C2semidirect
8. #11227C2semidirect
80. C2×#102Q8semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#19882
C2×#19252
#151442
#139182
#113842
#113222
#112562
#112272
#110762
#110662